The product of the digits in a two-digit number is 54. When 27 is added to the number the digits interchange their places.

(a) What does x represent in the two digit number 10x + y?

(b) Find the number.

(c) How much should be subtracted from the original number so that the ratio of new number to the original number will be 2; 3?

Let's break down and solve the problem step by step.

### (a) What does \( x \) represent in the two-digit number \( 10x + y \)?

In the expression \( 10x + y \), \( x \) represents the **tens digit** of the two-digit number.

### (b) Find the number.

Given conditions:

1. The product of the digits is 54: 
   \[
   x \times y = 54
   \]
2. Adding 27 to the number causes the digits to interchange their places:
   \[
   10x + y + 27 = 10y + x
   \]

Let's solve the second equation first. Rearranging terms:
\[
10x + y + 27 = 10y + x
\]
\[
10x - x + y - 10y = -27
\]
\[
9x - 9y = -27
\]
\[
x - y = -3
\]
So,
\[
x = y - 3
\]

Substitute \( x = y - 3 \) into the first equation:
\[
(y - 3) \times y = 54
\]
\[
y^2 - 3y = 54
\]
\[
y^2 - 3y - 54 = 0
\]

Now, solve this quadratic equation using the quadratic formula:
\[
y = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}
\]
where \( a = 1 \), \( b = -3 \), and \( c = -54 \):
\[
y = \frac{3 \pm \sqrt{(-3)^2 - 4(1)(-54)}}{2(1)}
\]
\[
y = \frac{3 \pm \sqrt{9 + 216}}{2}
\]
\[
y = \frac{3 \pm \sqrt{225}}{2}
\]
\[
y = \frac{3 \pm 15}{2}
\]

So, \( y = \frac{18}{2} = 9 \) or \( y = \frac{-12}{2} = -6 \) (which is not possible since \( y \) must be a digit).

Thus, \( y = 9 \).

Substituting \( y = 9 \) into \( x = y - 3 \):
\[
x = 9 - 3 = 6
\]

The two-digit number is \( 10x + y = 10(6) + 9 = 69 \).

### (c) How much should be subtracted from the original number so that the ratio of the new number to the original number will be \( \frac{2}{3} \)?

Let the original number be 69. Let the amount to be subtracted be \( k \).

The new number after subtraction is \( 69 - k \). The ratio of the new number to the original number is:
\[
\frac{69 - k}{69} = \frac{2}{3}
\]

Cross-multiply to solve for \( k \):
\[
3(69 - k) = 2 \times 69
\]
\[
207 - 3k = 138
\]
\[
207 - 138 = 3k
\]
\[
69 = 3k
\]
\[
k = 23
\]

Thus, \( 23 \) should be subtracted from 69 to obtain the required ratio.

### Summary:

1. \( x \) represents the tens digit of the two-digit number.
2. The number is 69.
3. The amount to be subtracted is 23.

Would you like further details on any part of this problem or have any questions?

Here are some related questions:
1. What is the significance of digit interchanging in such problems?
2. Can you solve similar problems with different products of digits?
3. How would the solution change if the difference between the digits was positive?
4. What would the original number be if the digits sum instead of their product was given?
5. How can you solve the quadratic equation using factorization?
6. What is the general approach to solving digit-based problems in algebra?
7. How would the problem change if 27 was subtracted instead of added to the number?
8. What other methods can be used to find the roots of a quadratic equation?

**Tip:** When solving digit-based problems, always consider how the digits and their operations relate to forming and solving equations.

Explore how to solve a mathematical problem involving a two-digit number where the product of its digits is 54 and adding 27 causes their digits to interchange. Learn step-by-step with detailed explanations and calculations.

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